Optimal. Leaf size=71 \[ -\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}-\frac {8 b \left (a-b x^4\right )^{3/4}}{77 a^2 x^7}-\frac {32 b^2 \left (a-b x^4\right )^{3/4}}{231 a^3 x^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {277, 270}
\begin {gather*} -\frac {32 b^2 \left (a-b x^4\right )^{3/4}}{231 a^3 x^3}-\frac {8 b \left (a-b x^4\right )^{3/4}}{77 a^2 x^7}-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^{12} \sqrt [4]{a-b x^4}} \, dx &=-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}+\frac {(8 b) \int \frac {1}{x^8 \sqrt [4]{a-b x^4}} \, dx}{11 a}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}-\frac {8 b \left (a-b x^4\right )^{3/4}}{77 a^2 x^7}+\frac {\left (32 b^2\right ) \int \frac {1}{x^4 \sqrt [4]{a-b x^4}} \, dx}{77 a^2}\\ &=-\frac {\left (a-b x^4\right )^{3/4}}{11 a x^{11}}-\frac {8 b \left (a-b x^4\right )^{3/4}}{77 a^2 x^7}-\frac {32 b^2 \left (a-b x^4\right )^{3/4}}{231 a^3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 43, normalized size = 0.61 \begin {gather*} \frac {\left (a-b x^4\right )^{3/4} \left (-21 a^2-24 a b x^4-32 b^2 x^8\right )}{231 a^3 x^{11}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 40, normalized size = 0.56
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{8}+24 a b \,x^{4}+21 a^{2}\right )}{231 a^{3} x^{11}}\) | \(40\) |
trager | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{8}+24 a b \,x^{4}+21 a^{2}\right )}{231 a^{3} x^{11}}\) | \(40\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{8}+24 a b \,x^{4}+21 a^{2}\right )}{231 a^{3} x^{11}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 55, normalized size = 0.77 \begin {gather*} -\frac {\frac {77 \, {\left (-b x^{4} + a\right )}^{\frac {3}{4}} b^{2}}{x^{3}} + \frac {66 \, {\left (-b x^{4} + a\right )}^{\frac {7}{4}} b}{x^{7}} + \frac {21 \, {\left (-b x^{4} + a\right )}^{\frac {11}{4}}}{x^{11}}}{231 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 39, normalized size = 0.55 \begin {gather*} -\frac {{\left (32 \, b^{2} x^{8} + 24 \, a b x^{4} + 21 \, a^{2}\right )} {\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{231 \, a^{3} x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.97, size = 1068, normalized size = 15.04 \begin {gather*} \begin {cases} - \frac {21 a^{4} b^{\frac {19}{4}} \left (\frac {a}{b x^{4}} - 1\right )^{\frac {3}{4}} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} + \frac {18 a^{3} b^{\frac {23}{4}} x^{4} \left (\frac {a}{b x^{4}} - 1\right )^{\frac {3}{4}} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} - \frac {5 a^{2} b^{\frac {27}{4}} x^{8} \left (\frac {a}{b x^{4}} - 1\right )^{\frac {3}{4}} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} + \frac {40 a b^{\frac {31}{4}} x^{12} \left (\frac {a}{b x^{4}} - 1\right )^{\frac {3}{4}} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} - \frac {32 b^{\frac {35}{4}} x^{16} \left (\frac {a}{b x^{4}} - 1\right )^{\frac {3}{4}} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {21 a^{4} b^{\frac {19}{4}} \left (- \frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} + \frac {18 a^{3} b^{\frac {23}{4}} x^{4} \left (- \frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} - \frac {5 a^{2} b^{\frac {27}{4}} x^{8} \left (- \frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} + \frac {40 a b^{\frac {31}{4}} x^{12} \left (- \frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} - \frac {32 b^{\frac {35}{4}} x^{16} \left (- \frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{64 a^{5} b^{4} x^{8} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) - 128 a^{4} b^{5} x^{12} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 64 a^{3} b^{6} x^{16} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.26, size = 59, normalized size = 0.83 \begin {gather*} -\frac {21\,a^2\,{\left (a-b\,x^4\right )}^{3/4}+32\,b^2\,x^8\,{\left (a-b\,x^4\right )}^{3/4}+24\,a\,b\,x^4\,{\left (a-b\,x^4\right )}^{3/4}}{231\,a^3\,x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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